If `alpha, beta` are the roots of the equation `2x^(2)-3x+2=0`, form the equation whose roots are `alpha^(2), beta^(2)`.

Answer

To solve the problem of forming an equation whose roots are \( \alpha^2 \) and \( \beta^2 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 - 3x + 2 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the given quadratic equation
The given equation is:
\[ 2x^2 - 3x + 2 = 0 \]
Here, \( A = 2 \), \( B = -3 \), and \( C = 2 \).

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{-3}{2} = \frac{3}{2} \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{C}{A} = \frac{2}{2} = 1 \]

### Step 3: Find \( \alpha^2 + \beta^2 \) and \( \alpha^2 \beta^2 \)
To find the new roots \( \alpha^2 \) and \( \beta^2 \), we need to calculate:
- The sum of the squares of the roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2 \cdot 1 = \frac{9}{4} - 2 = \frac{9}{4} - \frac{8}{4} = \frac{1}{4} \]

- The product of the squares of the roots:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = 1^2 = 1 \]

### Step 4: Form the new quadratic equation
The new quadratic equation with roots \( \alpha^2 \) and \( \beta^2 \) can be expressed as:
\[ x^2 - (\alpha^2 + \beta^2)x + \alpha^2 \beta^2 = 0 \]
Substituting the values we calculated:
\[ x^2 - \left(\frac{1}{4}\right)x + 1 = 0 \]

### Step 5: Multiply through by 4 to eliminate the fraction
To eliminate the fraction, we multiply the entire equation by 4:
\[ 4x^2 - x + 4 = 0 \]

### Final Answer
The equation whose roots are \( \alpha^2 \) and \( \beta^2 \) is:
\[ 4x^2 - x + 4 = 0 \]

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